Question

let f(u) = u⁴ and g(x) = u = 3x⁴ + 1. find (f ∘ g)(1).

(f ∘ g)(1) =
(type an exact answer.)

Explanation:

Step1: Apply Chain Rule

The chain rule states that \((f \circ g)'(x) = f'(g(x)) \cdot g'(x)\). First, find \(f'(u)\) and \(g'(x)\).
For \(f(u) = u^4\), \(f'(u) = 4u^3\) (using power rule \(\frac{d}{du}u^n = nu^{n - 1}\)).
For \(g(x)=3x^4 + 1\), \(g'(x)=12x^3\) (power rule: \(\frac{d}{dx}(3x^4)=12x^3\), derivative of constant 1 is 0).

Step2: Substitute \(g(x)\) into \(f'\)

\(f'(g(x)) = 4(3x^4 + 1)^3\) (substitute \(u = g(x)=3x^4 + 1\) into \(f'(u)\)).

Step3: Compute \((f \circ g)'(x)\)

Multiply \(f'(g(x))\) and \(g'(x)\): \((f \circ g)'(x)=4(3x^4 + 1)^3\cdot12x^3 = 48x^3(3x^4 + 1)^3\).

Step4: Evaluate at \(x = 1\)

Substitute \(x = 1\) into \((f \circ g)'(x)\):
First, \(3(1)^4+1 = 3 + 1 = 4\).
Then, \(48(1)^3(4)^3=48\times1\times64\) (since \(4^3 = 64\) and \(1^3 = 1\)).
Calculate \(48\times64 = 3072\).

Answer:

\(3072\)